# American Institute of Mathematical Sciences

August  2013, 33(8): 3329-3353. doi: 10.3934/dcds.2013.33.3329

## On the control of non holonomic systems by active constraints

 1 Department of Mathematics, Penn State University, University Park, Pa.16802 2 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, United States 3 Dipartimento di Matematica Pura ed Applicata, Università di Padova, Padova 35141, Italy

Received  April 2012 Revised  August 2012 Published  January 2013

The paper is concerned with mechanical systems which are controlled by implementing a number of time-dependent, frictionless holonomic constraints. The main novelty is due to the presence of additional non-holonomic constraints. We develop a general framework to analyze these problems, deriving the equations of motion and studying the continuity properties of the "control-to-trajectory" maps. Various geometric characterizations are provided, in order that the equations be affine w.r.t. the time derivative of the control. In this case the system is fit for jumps, and the evolution is well defined also in connection with discontinuous control functions. The classical Roller Racer provides an example where the non-affine dependence of the equations on the derivative of the control is due only to the non-holonomic constraint. This is a case where the presence of quadratic terms in the equations can be used for controllability purposes.
Citation: Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329
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